Start with the equation \(\ln(x+2) = 2 + \ln x\).

Use the properties of logarithms to combine the logarithms: \(\ln(x+2) - \ln x = 2\).

This simplifies to \(\ln\left(\frac{x+2}{x}\right) = 2\).

Exponentiate both sides to remove the logarithm: \(\frac{x+2}{x} = e^2\).

Multiply both sides by \(x\) to get \(x+2 = e^2 x\).

Rearrange to solve for \(x\): \(x - e^2 x = -2\).

Factor out \(x\): \(x(1 - e^2) = -2\).

Solve for \(x\): \(x = \frac{-2}{1 - e^2}\).

Calculate \(x\) to three decimal places: \(x \approx 0.313\).

Start with the equation \(\ln(1 + x) = 1 + \ln x\).

Use the property of logarithms: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).

Rearrange the equation: \(\ln(1 + x) - \ln x = 1\).

This becomes \(\ln \left( \frac{1 + x}{x} \right) = 1\).

Exponentiate both sides to remove the logarithm: \(\frac{1 + x}{x} = e\).

Rearrange to solve for \(x\): \(1 + x = ex\).

\(1 = ex - x\).

\(1 = x(e - 1)\).

\(x = \frac{1}{e - 1}\).

Calculate \(x\) using \(e \approx 2.718\):

\(x \approx \frac{1}{2.718 - 1} \approx 0.582\).

Round to 2 significant figures: \(x = 0.58\).

Start with the equation:

\(\ln(2x - 1) = 2 \ln(x + 1) - \ln x\)

Apply the logarithm power rule: \(2 \ln(x + 1) = \ln((x + 1)^2)\)

So the equation becomes:

\(\ln(2x - 1) = \ln((x + 1)^2) - \ln x\)

Use the logarithm quotient rule: \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\)

\(\ln(2x - 1) = \ln\left(\frac{(x + 1)^2}{x}\right)\)

Since the logarithms are equal, the arguments must be equal:

\(2x - 1 = \frac{(x + 1)^2}{x}\)

Multiply both sides by \(x\) to clear the fraction:

\(x(2x - 1) = (x + 1)^2\)

Expand both sides:

\(2x^2 - x = x^2 + 2x + 1\)

Rearrange to form a quadratic equation:

\(2x^2 - x - x^2 - 2x - 1 = 0\)

\(x^2 - 3x - 1 = 0\)

Solve the quadratic equation using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a = 1\), \(b = -3\), \(c = -1\)

\(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}\)

\(x = \frac{3 \pm \sqrt{9 + 4}}{2}\)

\(x = \frac{3 \pm \sqrt{13}}{2}\)

Calculate the positive root:

\(x = \frac{3 + \sqrt{13}}{2} \approx 3.303\)

Start by taking the natural logarithm of both sides of the equation:

\(\ln(2^{3x-1}) = \ln(5 \cdot 3^{1-x})\)

Apply the logarithm power rule: \(\ln(a^b) = b \ln a\).

\((3x-1)\ln 2 = \ln 5 + (1-x)\ln 3\)

Expand and rearrange the equation:

\(3x \ln 2 - \ln 2 = \ln 5 + \ln 3 - x \ln 3\)

Combine like terms:

\(3x \ln 2 + x \ln 3 = \ln 5 + \ln 3 + \ln 2\)

Factor out \(x\):

\(x(3 \ln 2 + \ln 3) = \ln 5 + \ln 3 + \ln 2\)

Solve for \(x\):

\(x = \frac{\ln 5 + \ln 3 + \ln 2}{3 \ln 2 + \ln 3}\)

Combine the logarithms:

\(x = \frac{\ln(5 \cdot 3 \cdot 2)}{\ln(2^3 \cdot 3)}\)

\(x = \frac{\ln 30}{\ln 24}\)

Start with the equation \(2^{3x-1} = 5 \cdot 3^{-x}\).

Take the natural logarithm of both sides: \(\ln(2^{3x-1}) = \ln(5 \cdot 3^{-x})\).

Apply the logarithm laws: \((3x-1)\ln 2 = \ln 5 + (-x)\ln 3\).

Rearrange to form a linear equation: \(3x \ln 2 - x \ln 3 = \ln 5 + \ln 2\).

Factor out \(x\): \(x(3 \ln 2 - \ln 3) = \ln 5 + \ln 2\).

Solve for \(x\): \(x = \frac{\ln 5 + \ln 2}{3 \ln 2 - \ln 3}\).

Simplify the expression: \(x = \frac{\ln 10}{\ln 24}\).